1. Recall that the [Lorentz group](https://en.wikipedia.org/wiki/Lorentz_group) $G=O(3,1)$ has 4 connected components
$$ G~=~G_0 ~\cup~ P\cdot G_0 ~\cup~ T\cdot G_0 ~\cup~ PT\cdot G_0. $$ 
Here the [connected components $G_0$ that contains the identity](https://en.wikipedia.org/wiki/Identity_component) is the restricted Lorentz group $G_0=SO^+(3,1)$.

2. It is straightforward to see that $G_0$ is a [normal subgroup](https://en.wikipedia.org/wiki/Normal_subgroup) of $G$. Therefore the [quotient](https://en.wikipedia.org/wiki/Quotient_group) $G/G_0$ is a group. 

3. For the Lorentz group $G=O(3,1)$, the quotient $G/G_0$ is isomorphic to the [Klein Vierergruppe](https://en.wikipedia.org/wiki/Klein_four-group) $V\cong \mathbb{Z}_2\times \mathbb{Z}_2$, which is generated by two elements.

4. Moreover, the quotient $G/G_0$ is discrete. Formally speaking, the elements of $G/G_0$ constitute the 'discrete Lorentz transformations'. In practice, one often picks a representative for each equivalence class, and called these representatives the 'discrete Lorentz transformations', with the implicit understanding that one might as well pick other representatives, that deviate with elements of $G_0$.    

5. Returning to OP's example, the difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, which belongs to $G_0$, cf. above comment by Meng Cheng.